Shred the cabbage, reserving half in a resealable plastic bag for another use. Heat the oil and butter in a large skillet over medium heat. Add the remaining cabbage, the pepper, and 1/2 teaspoon of the salt and sauté until soft and golden, 5 to 6 minutes. Add the peas and cook another minute. Remove from skillet to a plate, cover with foil, and set aside.
Add the chicken broth, lemon juice, and remaining salt to the skillet and bring to a simmer. Place the cod in the pan and poach, covered, until opaque, about 4 minutes total, flipping carefully once. (Cook in batches if necessary.) Remove the cod from the pan, reserving 3 fillets for another use.
Boil the poaching liquid over high heat until slightly thickened, about 2 minutes. Serve the cod over the cabbage, drizzling some of the cooking liquid over each fillet.
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